There are many interesting matrices associated to graphs. We all know about the adjacency matrix and the Laplacian, the basic matrices of spectral graph theory. The distance matrix is another interesting one - it was famously shown by Graham and Pollack that distance determinants of trees depend only on the number of vertices. The characteristic polynomial of distance matrices of trees was further studied by Graham and Lovasz, who found many interesting properties. Recently, graph theorists have begun to consider "exponential distance matrices" of graphs, obtained by taking the entrywise exponential of the usual distance matrix, and have proved some basic theorems on their eigenvalues for simple families of graphs. Taking a less myopic view of the mathematical landscape quickly reveals that exponential distance matrices appeared some thirty years ago in quantum physics, when Zagier explicitly evaluated the determinant of the exponential distance matrix of the Coxeter-Cayley graph of the symmetric group as the main step in proving the existence of a Hilbert space representation of deformed commutation relations interpolating between bosons and fermions. I will describe parallel results for the Hurwitz-Cayley graph of the symmetric group and explain their relation to gauge-string/dualities in Yang-Mills theory. As in Zagier's study, the main tools come from discrete harmonic analysis, aka the character theory of finite groups, and some basic aspects of symmetric function theory also play an important role. From a pedagogical perspective, the moral of the story is that it's good to imbibe some algebra with your combinatorics, and plain old matrices just don't cut it.    

This lecture concerns the metric Riemannian geometry of Einstein manifolds, which is a central theme in modern differential geometry and is deeply connected to a large variety of fundamental problems in algebraic geometry, geometric topology, analysis of nonlinear PDEs, and mathematical physics. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics.

My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. Such problems constitute the most challenging part in the metric geometry of Einstein manifolds. We will introduce recent major progress in the field. If time permits, I will propose several important open questions

Heuristically, two stochastic processes are dual if one can find a function to study one process by using the other. Implicitly, this technique can be traced back to the work of Blaise Pascal. Explicitly, it has been studied in different contexts, including interacting particle systems, and it is a crucial concept in population genetics.

Additionally, we will explore the duality between theoretical and applied mathematics. Specifically, we will examine instances in which theoretical probability is employed to study biological problems and situations where biological questions inspire interesting mathematical models. This discussion will encompass examples from population genetics, experimental evolution, and public health.

The Weierstrass p-function plays a great role in the classic theory of complex elliptic curves. A related function, the Weierstrass zeta-function, is used by Guerzhoy to construct preimages under the xi-operator of newforms of weight 2, corresponding to elliptic curves.  In this talk, I will discuss a generalization of the Weierstrass zeta-function and an application to harmonic Maass forms. More precisely, I will describe a construction of a preimage of the xi-operator of a newform of weight k for k>2. This is based on joint work with C. Alfes-Neumann, J. Funke and M. Mertens.

 Traditional numerical methods based on expensive matrix factorizations struggle with the scale of modern scientific applications. For example, kernel-based algorithms take a data set of size $N$, form the kernel matrix of size $N x N$, and then perform an eigendecomposition or inversion at a cost of $O(N^3)$ operations. For data sets of size $N \geq 10^5$, kernel learning is too expensive, straining the limits of personal workstations and even dedicated computing clusters. Randomized iterative methods have emerged as a faster alternative to the classical approaches. These methods combine randomized exploration with information about which matrix structures are important, leading to significant speed gains.

In this talk, I will review recent developments concerning two randomized algorithms. The first is "randomized block Krylov iteration", which uses an array of random Gaussian test vectors to probe a large data matrix in order to provide a randomized principal component analysis. Remarkably, this approach works well even when the matrix of interest is not low-rank. The second algorithm is "randomly pivoted Cholesky decomposition", which iteratively samples columns from a positive semidefinite matrix using a novelty metric and reconstructs the matrix from the randomly selected columns. Ultimately, both algorithms furnish a randomized approximation of an N x N matrix with a reduced rank $k << N$, which enables fast inversion or singular value decomposition at a cost of $O(N k^2)$ operations. The speed-up factor from $N^3$ to $N k^2$ operations can be 3 million. The newest algorithms achieve this speed-up factor while guaranteeing performance across a broad range of input matrices.

Given a fibration of complex projective manifolds $f:X\rightarrow Y$ with general fiber $F$, if the stable base locus of $-K_X $ is vertical then a theorem of Chang establishes the inequality $\kappa(-K_X)\leq \kappa(-K_Y) +\kappa(-K_F)$. In this talk I am going to discuss a generalization of this result to fibrations in positive characteristic satisfying certain tameness conditions. This is based on a joint project with Marta Benozzo and Chi-Kang Chang.

Matroids combinatorially abstract the ubiquitous notion of "independence" in various contexts such as linear algebra and graph theory.  Recently, an algebro-geometric perspective known as "combinatorial Hodge theory" led by June Huh produced several breakthroughs in matroid theory.  We first give an introduction to matroid theory in this light.  Then, we introduce a new geometric model for matroids that unifies, recovers, and extends various results from previous geometric models of matroids.  We conclude with a glimpse of new questions that further probe the boundary between combinatorics and algebraic geometry.  Joint works with Andrew Berget, Alex Fink, June Huh, Matt Larson, Hunter Spink, and Dennis Tseng.